# Velocity of a rocket in the earth moon system

#### s_gunn

Hi all! I've just finished writing a program in python that determines the position of a rocket in the earth moon system. Initially it is in circular orbit around the earth and then it is given a velocity boost at some point in it's orbit to make it travel to the moon.

I started all 3 bodies on the x axis with the earth at the left of the system's centre of mass (COM) and the moon at the right. the rocket is between the earth and moon and therefore lies to the right of the COM and all travel anti-clockwise.

I devised the initial velocity of the rocket using v=sqrt(G*mass of earth/radius of rocket) but it's just occured to me that it maybe should have been adjusted slightly as the earth has a small rotation in the same direction as the rocket around the system's COM.

Should the earth's angular velocity be added (or taken away) from the initial velocity of the rocket in orbit? and does the moon affect the initial velocity?

It's been puzzling me all day so any comments would be appreciated!

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#### tony873004

Gold Member
The Earth's rotation helped you when you launched your rocket. It gave you a head start. But once in orbit, your circular velocity formula is not affected by the Earth's rotation. The Moon is a small perturbation to your rocket in low-earth orbit, but not very significant. You can safely ignore it. That's because the Moon pulls the Earth with about the same strength it pulls your rocket, so the effect is negated.

#### s_gunn

I can't get it into my head why it wouldn't affect the rocket's orbital velocity as the path of the orbit can't be circular around the earth-moon system's centre of mass with the COM being about 3/4 way to the edge of the earth's radius!! Surely, the rocket must spiral slightly around the path of the earth's orbit?

When my uni demonstrator looked at my code, he added in a negative term of 2*pi*Earth-COM distance/Time Period of the Earth to my velocity and never explained why. I was hoping I'd come up with a reason but I also thought that it should have been a positive contribution! I also have a friend who did the same project but he added a contribution from the angular velocity from the moon which i couldn't determine a reason for either.

Any ideas?

#### K^2

What's your initial CoM velocity for the rocket?

If rocket starts in Earth orbit, the Moon provides only minor perturbation. It can be ignored.

Maybe you can post the code? I could tell a lot more from that than from your explanation of what it is supposed to do.

#### s_gunn

i had to convert it from python to a word doc so i hope the indents are still intact!! I'm also relatively new to programming so forgive my messy set-up, the code wasn't marked just the report we've written on it so I didn't try making it 'pretty'!

I hope you can make sense of it!

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#### tony873004

Gold Member
Your spaceship in Low Earth Orbit (LEO) is not orbiting the COM of the Earth/Moon system, as it is orbiting interior to the Moon's orbit. It is orbiting the Earth only with minor perturbations from the Moon.

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